A Bilevel Approach to the Facility Location Problem with Customer Preferences Under a Mill Pricing Policy
This paper addresses the facility location problem under a mill pricing policy, integrating customers’ behavior through the concept of preferences. The problem is modeled as a bilevel optimization problem, where the existence of ties in customers’ preferences can lead to an ill-posed bilevel problem...
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MDPI AG
2024-11-01
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| Series: | Mathematics |
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| author | Herminia I. Calvete Carmen Galé Aitor Hernández José A. Iranzo |
| author_facet | Herminia I. Calvete Carmen Galé Aitor Hernández José A. Iranzo |
| author_sort | Herminia I. Calvete |
| collection | DOAJ |
| description | This paper addresses the facility location problem under a mill pricing policy, integrating customers’ behavior through the concept of preferences. The problem is modeled as a bilevel optimization problem, where the existence of ties in customers’ preferences can lead to an ill-posed bilevel problem due to the possible existence of multiple optima to the lower-level problem. As the commonly employed optimistic and pessimistic strategies are inadequate for this problem, a specific approach is proposed bearing in mind the customers’ rational behavior. In this work, we propose a novel formulation of the problem as a bilevel model in which each customer faces a lexicographic biobjective problem in which the preference is maximized and the total cost of accessing the selected facility is minimized. This allows for a more accurate representation of customer preferences and the resulting decisions regarding facility location and pricing. To address the complexities of this model, we apply duality theory to the lower-level problems and, ultimately, reformulate the bilevel problem as a single-level mixed-integer optimization problem. This reformulation incorporates big-<i>M</i> constants, for which we provide valid bounds to ensure computational tractability and solution quality. The computational study conducted allows us to assess, on the one hand, the effectiveness of the proposed reformulation to address the bilevel model and, on the other hand, the impact of the length of the customer preference lists and fixed opening cost for facilities on the computational time and the optimal solution. |
| format | Article |
| id | doaj-art-bebe155f3b3e40f4b8b30d906a6fb424 |
| institution | OA Journals |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj-art-bebe155f3b3e40f4b8b30d906a6fb4242025-08-20T02:04:54ZengMDPI AGMathematics2227-73902024-11-011222345910.3390/math12223459A Bilevel Approach to the Facility Location Problem with Customer Preferences Under a Mill Pricing PolicyHerminia I. Calvete0Carmen Galé1Aitor Hernández2José A. Iranzo3Departamento de Métodos Estadísticos, Instituto Universitario de Matemáticas y Aplicaciones (IUMA), Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, SpainDepartamento de Métodos Estadísticos, Instituto Universitario de Matemáticas y Aplicaciones (IUMA), Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, SpainDepartamento de Métodos Estadísticos, Instituto Universitario de Matemáticas y Aplicaciones (IUMA), Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, SpainDepartamento de Métodos Estadísticos, Instituto Universitario de Matemáticas y Aplicaciones (IUMA), Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, SpainThis paper addresses the facility location problem under a mill pricing policy, integrating customers’ behavior through the concept of preferences. The problem is modeled as a bilevel optimization problem, where the existence of ties in customers’ preferences can lead to an ill-posed bilevel problem due to the possible existence of multiple optima to the lower-level problem. As the commonly employed optimistic and pessimistic strategies are inadequate for this problem, a specific approach is proposed bearing in mind the customers’ rational behavior. In this work, we propose a novel formulation of the problem as a bilevel model in which each customer faces a lexicographic biobjective problem in which the preference is maximized and the total cost of accessing the selected facility is minimized. This allows for a more accurate representation of customer preferences and the resulting decisions regarding facility location and pricing. To address the complexities of this model, we apply duality theory to the lower-level problems and, ultimately, reformulate the bilevel problem as a single-level mixed-integer optimization problem. This reformulation incorporates big-<i>M</i> constants, for which we provide valid bounds to ensure computational tractability and solution quality. The computational study conducted allows us to assess, on the one hand, the effectiveness of the proposed reformulation to address the bilevel model and, on the other hand, the impact of the length of the customer preference lists and fixed opening cost for facilities on the computational time and the optimal solution.https://www.mdpi.com/2227-7390/12/22/3459facility locationmill pricingpreferencesbilevel optimizationlexicographic biobjective |
| spellingShingle | Herminia I. Calvete Carmen Galé Aitor Hernández José A. Iranzo A Bilevel Approach to the Facility Location Problem with Customer Preferences Under a Mill Pricing Policy Mathematics facility location mill pricing preferences bilevel optimization lexicographic biobjective |
| title | A Bilevel Approach to the Facility Location Problem with Customer Preferences Under a Mill Pricing Policy |
| title_full | A Bilevel Approach to the Facility Location Problem with Customer Preferences Under a Mill Pricing Policy |
| title_fullStr | A Bilevel Approach to the Facility Location Problem with Customer Preferences Under a Mill Pricing Policy |
| title_full_unstemmed | A Bilevel Approach to the Facility Location Problem with Customer Preferences Under a Mill Pricing Policy |
| title_short | A Bilevel Approach to the Facility Location Problem with Customer Preferences Under a Mill Pricing Policy |
| title_sort | bilevel approach to the facility location problem with customer preferences under a mill pricing policy |
| topic | facility location mill pricing preferences bilevel optimization lexicographic biobjective |
| url | https://www.mdpi.com/2227-7390/12/22/3459 |
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